I am trying to see if its possible to toss all the computers in the server room and just run cables out of it.

The end goal is you just sit down at your desk and all you have is a display and peripherals. No computer can be seen or heard.

One idea is just to pass the required cables through the wall of the server room and setup the display and peripherals on the other side. But how do you do it in such a way you don't get all the server room noise? The required cables are different for every user, you can't just pass exactly what you need and seal it all up.

Another idea is just to use Ethernet for everything. You can get DVI/audio/USB over Ethernet. But then that's super expensive.

EDIT

And these computers will be used for more than just email. They will be powerful and noisy machines.

If you are looking to take existing systems and move them into a separate room, that is straight-forward, although likely hard to execute. You buy devices like this which convert KVM cables to Cat5, and patch the desk location to the computer location. The difficulty is more in where you put the computers and how you manage the cooling, power and wire. This will not scale well.NOTE: if physical security is the concern, it might be easier to use a locking box (like this) for the CPU units.

If you are looking to update the infrastructure, there are two approachs.

One is "desktop virtualization" (aka terminal services). Basically you put a small number of servers up, setup terminal services (or Citrix MetaFrame) on the servers, and deploy thin clients at each desk. The thin client only runs the program to connect to the terminal server, and the terminal server runs all of the desktop applications.NOTE: PanoLogic has a very interesting desktop virtualization solution.

The second is to implement a "Blade PC" or "Centralized Desktop" solution (like ClearCube) where you install specialized computers and hardware in the data center, and special connector boxes at the desktop. Conceptually similar to the "use your own PC" method, but it scales well, works well, but is somewhat costly.

We have several stand-alone servers with no monitor, keyboard, mouse - just a machine in a room. We access them with Remote Desktop. It is very convenient and isnt any different than sitting right in front of the machine.

You could cheat and get thin clients that mount on the back of monitors (called a 'VESA mount') and use those to run remote desktop/vnc fullscreen to the 'big iron' that you keep in the closet. Note that this will work fine for most dev work, but things requiring fast graphics (ie. playing quake or lots of fly-though rendering while working on 3D modelling or animation) will of course suffer.

My goal here is to reduce noise,

There are fanless VESA-mounted machines.

reduce cleaning (of dust off computers),

It's "part of the monitor" more or less, and small at that

reduce cooling by centralizing and removing computer cases,

The big iron that requires massive cooling is in the closet

save space,

Being VESA-mounted, they're on the back of your monitor, not on your desktop

and increase security.

Remote desktops require a login to get to the big iron where the real work is done, so walking off with one doesn't get you anything beyond the physical hardware (of which the display is almost as expensive as the computer).